Calculus I Midterm 2 Practice Problems 1. Use the linear approximation of the function f (x) = (x + 1)1/4 to estimate (1.02)1/4.
2. Two hallways, one 8 feet wide and the other 1 feet wide, meet at right angles. Determine the length of the longest ladder that can be carried horizontally from one hallway to the other.
3. Find the point on the curve y = x2 closest to the point (0, 1)
4. Suppose a wire 4 ft long is to be cut into two pieces. One will be formed into a square and the other one will be formed into a regular triangle. Find the size of each piece to minimize the total area of the two region.
5. Suppose a 6-ft tall person is 12 ft away from an 18-ft tall lamppost. If the person is moving away from the lamppost at a rate of 2 ft/s, at what rate is the length of the shadow changing?
6. Parametric equations for the position of an object is given. Find the object’s velocity and speed at the given times, and describe its motion.
x= 2 cos 2t + sin 5t
y= 2 sin 2t + cos 5t (a)t = 0, (b)t = π 2 7. Find the derivatice of the function
f(x) = Z cos x
x
p1 − t2dt
8. If x sin x =Rx2
0 f(t) dt, where f is a continuous function, find f (4).
9. R 1
x2−4x+3dx 10. R 1
x2−4x+5dx 11. R ln x
x dx 12. R e2xsin 2x dx 13. R2
0 x2 (x2+4)2 dx 14. R 1
√x+xdx 15. R tan4xsec4x dx 16. R tan x dx 17. R sec x dx 18. R1
−2|2x + 1| dx 19. R1
0 x−1/3dx 20. R∞
1 x−1/3dx 21. R∞
0 cos x dx 22. R∞
1
sin x+2 x dx
23. Find the average value of f (x) =√
x on the interval [0, 9]
24. Find the area of the region bounded by y = 2 cos x, y = sin 2x for x ∈ [−π, π]
25. Let Ω be the region bounded by y = sec x, x = 0, x = π4 and y = 0. Find integrals represent the volume of the solids generated by Ω about (a) x-axis, (b) y-axis, (c) y = −1, (d) x = −1.
• Double-Angle
sin 2θ = 2 sin θ cos θ
cos 2θ = 2 cos
2θ − 1 = 1 − 2 sin
2θ
• Derivative formulas
d
dx
sin
−1x = 1
√ 1 − x
2,
d
dx
cos
−1x = − 1
√ 1 − x
2,
d
dx
tan
−1x = 1
1 + x
2,
d
dx
cot
−1x = − 1 1 − x
2,
d
dx
sec
−1x = 1
|x| √
x
2− 1 ,
d
dx
